A.13 Negative binomial distribution

X has a negative binomial distribution of index n and parameter π, denoted

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if it has a discrete distribution with density

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Because

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we sometimes use the notation

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The mean and variance are

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Because

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we see that p(X+1)> p(X) if and only if

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and hence that a mode occurs at

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the square brackets denoting ‘integer part of’, and this mode is unique unless  is an integer.

It can be shown that the distribution function can be found in terms of that of the binomial distribution, or equivalently in terms of the incomplete beta function; for details see Balakrishnan et al. (1992, Chapter 5, Section 6). Just as the Poisson distribution can arise as a limit of the binomial distribution, so it can as a limit of the negative binomial, but in this case as

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The particular case where n=1, so that  , is sometimes referred to as the geometric distribution.

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